Least-Squares State Estimation, LQR and LQ-Tracking

Abstract

This note is a tutorial on the Least-Squares State Estimator (LSSE) (the deterministic version of the Kalman-Bucy filter) and related topics. The LSSE is formulated as finding the state trajectory consistent with the system's equations with the minimal amount of L2 process and measurement uncertainty. As stated, this is an input-signal design problem with linear dynamics and affine-quadratic objective in the state and inputs, and therefore a deterministic optimal control problem. We explore its relations to other problems such as the Linear Quadratic Regulator (LQR) with initial or final conditions, as well as the Linear Quadratic (LQ)-tracking problem. Several related topics such as the use of homogeneous coordinates and time reversal in optimal control are explored. The emergence of dynamical controllers/estimators in both LQ-tracking and LSSE as opposed to memoryless ones (as in LQR) is highlighted. It is seen to be a consequence of the affine-quadratic, rather than a purely quadratic form of the cost objective. The relations with the stochastic version of the Kalman-Bucy filter are explicitly highlighted, as well as characterizations in terms of certainty (information) matrices, versus covariance matrices.